Convergence Rates of Ergodic Limits and Approximate Solutions

✍ Scribed by S.Y. Shaw

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 364 KB
• Volume: 75
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1095

✦ Synopsis

This paper is concerned with the convergence rates of two processes (\left{A_{x}\right}) and (\left{B_{x}\right}), under the assumption that (\left|A_{x}\right|=O(1)) and there is a closed operator (A) such that (B_{x} A \subset A B_{x}=I-A_{x},\left|A A_{x}\right|=O(e(\alpha))), and (B_{x}^{} x^{}=\varphi(\alpha) x^{}) for (x^{} \in R(A)^{\perp}), where (e(\alpha) \rightarrow 0) and (|\varphi(x)| \rightarrow \infty). It was previously proved that (\left{A_{x}\right}) converges strongly on (N(A) \oplus \overline{R(A)}) to (P), the projection onto (N(A)) along (\overline{R(A)}), and (\left{B_{x}\right}) converges strongly on (A(D(A) \cap \overline{R(A)})) to (A_{1}^{-1}), the inverse operator of (A_{1}=A \mid \widehat{R(A)}). In this paper, the two processes are shown to be saturated with order (O(e(\alpha))), and their saturation classes are characterized. The result provides a unified approach to convergence rates for many particular mean ergodic theorems and for various methods of solving the equation (A x=y). We discuss in particular applications to integrated semigroups, cosine operator functions, and tensor product semigroups. 1993 Academic Press, Inc.